Let's look at an example. Consider the rings $(R, +, *)$ and $(\mathbb{C}, +, *)$ of real and complex numbers respectively. We will show that $\mathbb{R}$ and $\mathbb{C}$ are NOT isomorphic.

To show this, suppose the opposite, i.e., suppose that $\mathbb{R}$ is isomorphic to $\mathbb{C}$. Then there exists an isomorphism, $\phi : \mathbb{C} \to \mathbb{R}$. Consider the element $i \in \mathbb{C}$. Then:

But $\phi (i) \in \mathbb{R}$. The equation above saids that a real number squared is equal to $-1$. This is a contradiction. So the assumption that $\mathbb{R}$ was isomorphic to $\mathbb{C}$ is false! Hence: